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If $\omega$ is an imaginary cube root of unity, then the value of the determinant $\left|\begin{array}{ccc}1+\omega & 0 & -\omega \\ 1+\omega^{2} & \omega & -\omega^{2} \\ \omega+\omega^{2} & \omega & -\omega^{2}\end{array}\right|$ is
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$-3 w^{2}$
$\left|\begin{array}{ccc}1+\omega & \omega^{2} & -\omega \\ 1+\omega^{2} & \omega & -\omega^{2} \\ \omega+\omega^{2} & \omega & -\omega^{2}\end{array}\right|$
$\left|\begin{array}{ccc}0 & \omega^{2} & -\omega \\ 0 & \omega & -\omega^{2} \\ -1+\omega & \omega & -\omega^{2}\end{array}\right|$
$=(-1+\omega)\left(-\omega^{4}+\omega^{2}\right)=(\omega-1)\left(\omega^{2}-c\right.$
$=\omega^{3}-\omega^{2}-\omega^{2}+\omega=-3 \omega^{2}$
$\left|\begin{array}{ccc}0 & \omega^{2} & -\omega \\ 0 & \omega & -\omega^{2} \\ -1+\omega & \omega & -\omega^{2}\end{array}\right|$
$=(-1+\omega)\left(-\omega^{4}+\omega^{2}\right)=(\omega-1)\left(\omega^{2}-c\right.$
$=\omega^{3}-\omega^{2}-\omega^{2}+\omega=-3 \omega^{2}$
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